A mathematical and statistical analysis of the -3/2 power rule of self-thinning in even-aged plant populations

The self-thinning rule for even-aged plant populations (also called the -3/2 power law or Yoda's law) is reviewed. This widely accepted but poorly understood generalization predicts that, through time, growth and mortality in a crowded population trace a straight thinning line of slope -3/2 in a log-log plot of average plant weight versus plant density. The evidence for this rule is examined, then reanalyzed to objectively evaluate the strength of support for the rule. Mathematical models are constructed to produce testable predictions about causal factors.

Major problems in the evidence for the thinning rule include inattention to contradictory data, lack of hypothesis testing, inappropriate curve-fitting techniques, and the use of an invalid data transformation. When these problems are corrected, many data sets thought to corroborate the rule do not demonstrate any size-density relationship. Also, the variations among thinning slopes and intercepts are much greater than currently accepted, many slopes disagree quantitatively with the thinning rule, and thinning slope and intercept differ among plant groups.

The models predict that thinning line slope is determined by the allometry between area occupied and plant weight, while the intercept is also related to the density of biomass per unit of space occupied and the partitioning of resources among competing individuals. Statistical tests confirm that thinning slope is correlated with several measures of plant allometry and that variations in thinning slope among plant groups reflect allometric differences.

The ultimate thinning line, which describes the overall size-density relationship among populations of many species, is a trivial geometric consequence of packing objects onto a surface. This cause differs from the factors positioning the self-thinning lines of individual populations, so the existence of an overall relationship is not relevant to the thinning rule.

The evidence does not support acceptance of the self-thinning rule as a quantitative biological law. The slopes and intercepts of size-density relationships are variable, and the slopes can be explained by simple geometric arguments.